Exponents and Logarithms

Exponents are a shorthand for repeated multiplication, but they quickly grow into one of the most powerful ideas in all of mathematics: a way to describe anything that grows or shrinks by a fixed factor over each step — money earning interest, bacteria doubling, radioactive atoms decaying, sound intensity climbing. Logarithms are the mirror image: they answer the question "what power do I need?" and, in doing so, turn impossible-looking multiplication into simple addition.

If exponents and logarithms feel like two separate topics you were forced to memorize, this page is here to fix that. They are a single idea seen from two directions — inverse operations, like squaring and square-rooting — and once you see them that way, pH, earthquakes, sound, radioactive dating, and the interest on your savings account all start to look like the same piece of algebra.

Learning Objectives

By the end of this page, you should be able to:

• Apply the laws of exponents fluently, including negative, zero, and fractional exponents.
• Explain what fractional exponents mean and connect them to roots.
• Model exponential growth and decay and interpret the base and rate.
• Define a logarithm as the inverse of an exponential and convert between the two forms.
• Use the log laws to expand, condense, and solve equations.
• Recognize and work with real logarithmic scales: pH, decibels, the Richter scale, and compound interest.

Quick Answer

An exponent tells you how many times to multiply a base by itself: $a^n = a \times a \times \cdots \times a$ ($n$ factors). The laws of exponents ($a^m a^n = a^{m+n}$, $(a^m)^n = a^{mn}$, $a^{-n} = 1/a^n$, $a^{1/n} = \sqrt[n]{a}$) all follow from that one idea. An exponential function $y = a \cdot b^x$ grows (if $b > 1$) or decays (if $0 < b < 1$) by a constant factor per step. A logarithm is the inverse: $\log_b y = x$ means exactly $b^x = y$ - it asks "what exponent produces $y$?" The log laws turn multiplication into addition ($\log(MN) = \log M + \log N$), which is why logarithms were invented. Real logarithmic scales (pH, decibels, Richter) compress enormous ranges of numbers into manageable ones.

Where It Came From

By the late 1500s, European astronomers and navigators faced a brutal bottleneck: multiplying and dividing the long, many-digit numbers that came out of astronomical tables and trigonometric calculations. A single multiplication of two seven-digit numbers, done by hand, could take a careful person the better part of an hour — and a voyage or a planetary prediction demanded thousands of them. Errors crept in constantly. The mathematics was known; the arithmetic was the enemy.

The Scottish landowner John Napier spent roughly twenty years attacking this problem, publishing his Mirifici Logarithmorum Canonis Descriptio ("Description of the Wonderful Canon of Logarithms") in 1614. His insight rested on a pattern people had noticed since antiquity: if you line up a geometric sequence (each term a fixed multiple of the last: $2, 4, 8, 16, 32, \dots$) against an arithmetic sequence (each term a fixed amount more: $1, 2, 3, 4, 5, \dots$), then multiplying two numbers in the top row corresponds to adding their partners in the bottom row. Napier's logarithms made this correspondence continuous and practical: to multiply two numbers, you looked up their logarithms, added them, and looked up the answer. Multiplication had been replaced by addition.

The English mathematician Henry Briggs travelled to Edinburgh to meet Napier and proposed a refinement: base the logarithms on 10 and set $\log 1 = 0$, giving the "common logarithms" that filled tables and slide rules for the next 350 years. Briggs laboriously computed logarithms to 14 decimal places for thousands of integers. The astronomer Johannes Kepler used logarithms to complete his planetary calculations; Pierre-Simon Laplace later said logarithms, "by shortening the labours, doubled the life of the astronomer." Only in the 1970s, when electronic calculators became cheap, did logarithm tables and slide rules finally retire — but the mathematics they revealed had by then become foundational to science.

The Laws of Exponents

Start from the definition. $a^n$ means $a$ multiplied by itself $n$ times, so $2^3 = 2 \times 2 \times 2 = 8$. Every exponent law is just bookkeeping of these factors.

Product rule: $a^m \cdot a^n = a^{m+n}$. Count factors: $a^2 \cdot a^3 = (a\,a)(a\,a\,a) = a^5$.

Quotient rule: $\dfrac{a^m}{a^n} = a^{m-n}$. Cancelling factors, $\dfrac{a^5}{a^2} = a^3$.

Power rule: $(a^m)^n = a^{mn}$, since $(a^2)^3 = a^2 \cdot a^2 \cdot a^2 = a^6$.

Products and quotients of bases: $(ab)^n = a^n b^n$ and $\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$.

Zero and negative exponents are forced on us by consistency. If the quotient rule is to hold when $m = n$, then $\dfrac{a^n}{a^n} = a^{0}$, and since anything divided by itself is $1$, we must define $a^0 = 1$ (for $a \ne 0$). Push one step further: $\dfrac{a^2}{a^3} = a^{-1}$, but that fraction equals $\dfrac{1}{a}$, so $a^{-n} = \dfrac{1}{a^n}$. Negative exponents are not "subtraction" — they mean reciprocal.

Fractional Exponents Are Roots

What could $a^{1/2}$ mean? Whatever it is, the power rule demands $\left(a^{1/2}\right)^2 = a^{(1/2)\cdot 2} = a^1 = a$. The number that squares to give $a$ is the square root, so $a^{1/2} = \sqrt{a}$. In general:

$a^{m/n} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m.$

Worked example. Simplify $\dfrac{16^{3/4} \cdot 2^{-1}}{8^{1/3}}$.

• $16^{3/4} = \left(\sqrt[4]{16}\right)^3 = 2^3 = 8$ (since $\sqrt[4]{16} = 2$).
• $2^{-1} = \dfrac{1}{2}$.
• $8^{1/3} = \sqrt[3]{8} = 2$.

So the expression is $\dfrac{8 \cdot \frac{1}{2}}{2} = \dfrac{4}{2} = 2$. Notice how mixing all three ideas — roots, negatives, and cancellation — reduces to a single clean number.

Exponential Growth and Decay

An exponential function has the variable in the exponent:

$y = a \cdot b^{x},$

where $a$ is the starting value (at $x = 0$) and $b$ is the constant growth factor per unit of $x$. If $b > 1$, $y$ grows; if $0 < b < 1$, $y$ decays. The defining feature is that $y$ changes by the same multiplicative factor over each equal step — unlike a linear function, which changes by the same additive amount.

Worked example — compound interest. You invest $1000 at 6% annual interest, compounded yearly. Each year the balance is multiplied by $1.06$, so after $t$ years:

$A = 1000 \cdot (1.06)^{t}.$

After 12 years: $A = 1000 \cdot (1.06)^{12}$. Since $(1.06)^{12} \approx 2.012$, you have about $2012 — your money has roughly doubled. This is the "Rule of 72" in action: $72 / 6 = 12$ years to double.

Worked example — decay. A radioactive sample has a half-life of 5 years, meaning it halves every 5 years. Starting from 80 grams, the amount after $t$ years is:

$m = 80 \cdot \left(\tfrac{1}{2}\right)^{t/5}.$

After 15 years, $t/5 = 3$, so $m = 80 \cdot (1/2)^3 = 80 / 8 = 10$ grams. Three half-lives, three halvings.

A special base, $e \approx 2.71828$, appears whenever growth is continuous rather than in discrete steps; continuously compounded interest gives $A = a\,e^{rt}$. We flag it here and explore it fully in calculus.

Logarithms: Exponents in Reverse

Here is the key sentence to memorize: a logarithm is an exponent. The statement

$\log_b y = x \quad\text{means exactly}\quad b^{x} = y.$

"$\log_b y$" reads as "the power you raise $b$ to in order to get $y$." So $\log_2 8 = 3$ because $2^3 = 8$, and $\log_{10} 1000 = 3$ because $10^3 = 1000$. Logarithm and exponential are inverse functions: each undoes the other. Two common shorthands: $\log x$ usually means $\log_{10} x$ (common log), and $\ln x$ means $\log_e x$ (natural log).

Because logs are exponents, the exponent laws become the log laws:

• $\log_b(MN) = \log_b M + \log_b N$ (products become sums — the original point of logarithms).
• $\log_b\!\left(\dfrac{M}{N}\right) = \log_b M - \log_b N$.
• $\log_b(M^p) = p\,\log_b M$ (this is what lets us pull an unknown exponent down).
• Change of base: $\log_b M = \dfrac{\log_c M}{\log_c b}$, letting you compute any log with a calculator's $\log$ or $\ln$ button.

Worked example — solving for an exponent. How long until the $1000 investment above triples? Solve $1000 \cdot (1.06)^t = 3000$, i.e. $(1.06)^t = 3$. Take the log of both sides and use the power law:

$t \,\log(1.06) = \log 3 \quad\Rightarrow\quad t = \frac{\log 3}{\log 1.06} = \frac{0.4771}{0.02531} \approx 18.85 \text{ years}.$

Without logarithms there is no clean way to get the exponent out of the exponent. This single move — take a log, drop the power down — is the workhorse of every growth-and-decay problem.

Real-World Applications

Logarithmic scales exist because nature spans ranges too vast for ordinary numbers. Compressing them with a log turns "billions to one" into "a few tens."

• pH (chemistry): $\text{pH} = -\log_{10}[\text{H}^+]$, where $[\text{H}^+]$ is the hydrogen-ion concentration in mol/L. Lemon juice at pH 2 is 100 000 times more acidic than water at pH 7 — a difference of 5 pH units means $10^5$ in concentration.
• Decibels (acoustics): loudness in dB is $10\log_{10}(I/I_0)$. Every 10 dB is a tenfold increase in intensity, which is why an 80 dB street and a 110 dB concert differ by a factor of $10^3 = 1000$ in energy.
• Richter / moment magnitude (seismology): each whole number up the scale is roughly $10\times$ the ground-shaking amplitude and about $32\times$ the energy. A magnitude 7 quake shakes 100 times harder than a magnitude 5.
• Finance: compound interest, loan amortization, and doubling times all rely on exponentials, and logs solve for time.
• Biology and medicine: bacterial growth, drug half-lives in the bloodstream, and epidemic case counts are exponential; carbon-14 dating inverts radioactive decay with a log.
• Computer science: the efficiency of binary search and balanced trees is $\log_2 n$ — doubling the data adds only one step.

Common Mistakes

Mistake 1: Thinking $a^{-n}$ is negative. Students read $2^{-3}$ as a negative number. It is not: $2^{-3} = \dfrac{1}{2^3} = \dfrac{1}{8}$, which is positive. A negative exponent flips to a reciprocal; it never changes the sign of the result.

Mistake 2: Splitting a log of a sum. It is tempting to write $\log(M + N) = \log M + \log N$. This is false. The real law is $\log(M \cdot N) = \log M + \log N$ — logs turn products into sums, never sums into sums. There is no simple rule for $\log(M+N)$; leave it as is.

Mistake 3: Adding exponents when multiplying different bases. The rule $a^m a^n = a^{m+n}$ needs the same base. So $2^3 \cdot 5^2$ is not $10^5$; it is $8 \cdot 25 = 200$. Only matching bases let you add exponents.

Mistake 4 (bonus): Mishandling the power inside a log. $\log_b(M^2)$ equals $2\log_b M$, but $(\log_b M)^2$ does not — the first squares the input, the second squares the whole logarithm. Watch where the exponent sits.

Comparison and Connections

Idea	Grows/changes by	Inverse of	Typical model
Linear	equal amounts added	subtraction	$y = mx + c$
Exponential	equal factors multiplied	logarithm	$y = a\,b^x$
Logarithmic	slows down as input grows	exponential	$y = \log_b x$
Power	base varies, exponent fixed	roots	$y = x^n$

The crucial contrast is exponential vs power: in $2^x$ the variable is the exponent; in $x^2$ the variable is the base. They behave completely differently — $2^x$ eventually overtakes any power $x^n$. And exponential vs logarithmic functions are reflections of each other across the line $y = x$, exactly because they are inverses.

Practice Questions

Recall

Evaluate $\log_3 81$ and $27^{2/3}$. Answer: $\log_3 81 = 4$ (since $3^4 = 81$); $27^{2/3} = \left(\sqrt[3]{27}\right)^2 = 3^2 = 9$.

Understanding

Explain why $a^0 = 1$ for any $a \ne 0$. Guidance: By the quotient rule, $\dfrac{a^n}{a^n} = a^{n-n} = a^0$. But any nonzero number divided by itself is $1$. For both to hold, $a^0$ must equal $1$.

Application

A town of 20 000 people grows 3% per year. Write a model and find the population after 25 years. Answer: $P = 20000 \cdot (1.03)^{25}$. Since $(1.03)^{25} \approx 2.094$, the population is about $41,875$ people.

Analysis

The sound at a concert is 1000 times more intense than normal conversation. How many decibels louder is it, and why does the answer feel "small"? Answer: $10\log_{10}(1000) = 10 \times 3 = 30$ dB louder. It feels small because the decibel scale is logarithmic: a huge multiplicative jump ($\times 1000$) becomes a modest additive one (+30 dB). That compression is exactly the purpose of the scale — it matches how our ears perceive loudness.

FAQ

Why do we even need logarithms now that calculators exist? The tables and slide rules are gone, but the function is essential: logarithms are the only clean way to solve for a variable stuck in an exponent, and log scales (pH, dB, Richter) are how science tames enormous ranges of data. The tool changed; the mathematics didn't.

What's the difference between $\log$ and $\ln$? $\log$ (with no base written) conventionally means base 10, the "common log." $\ln$ means base $e \approx 2.718$, the "natural log." They differ only by a constant factor, and either can compute the other via change of base.

Can I take the log of a negative number or zero? No — not with real numbers. Since $b^x$ is always positive for a positive base $b$, no exponent produces a negative result or zero, so $\log$ of a non-positive number is undefined in real algebra.

Why does $e$ keep showing up? $e$ is the base that describes continuous growth — growth compounded not yearly or monthly but every instant. It's also the base whose exponential function equals its own rate of change, which makes it the natural language of calculus.

How is a fractional exponent different from a negative one? They control different things. A fractional exponent takes a root ($a^{1/3} = \sqrt[3]{a}$); a negative exponent takes a reciprocal ($a^{-1} = 1/a$). A number like $a^{-1/2}$ does both: $\dfrac{1}{\sqrt{a}}$.

How do I know whether a situation is exponential or linear? Ask: does the quantity change by a fixed amount each step (linear, e.g. +$50 a month) or by a fixed percentage/factor (exponential, e.g. +5% a year)? "Per year," "doubles," "half-life," and "percent growth" almost always signal exponential.

Quick Revision

• $a^m a^n = a^{m+n}$, $\dfrac{a^m}{a^n} = a^{m-n}$, $(a^m)^n = a^{mn}$.
• $a^0 = 1$, $a^{-n} = \dfrac{1}{a^n}$, $a^{m/n} = \sqrt[n]{a^m}$.
• Exponential model: $y = a\,b^x$ — grows if $b > 1$, decays if $0 < b < 1$.
• Definition: $\log_b y = x \iff b^x = y$. A logarithm is an exponent.
• Log laws: $\log(MN) = \log M + \log N$; $\log(M/N) = \log M - \log N$; $\log(M^p) = p\log M$; change of base $\log_b M = \dfrac{\ln M}{\ln b}$.
• To free an exponent, take a log of both sides and use the power law.
• Log scales: pH $= -\log[\text{H}^+]$; decibels $= 10\log(I/I_0)$; each Richter unit $\approx \times 10$ amplitude.
• $\log$ of a sum does NOT split; only products/quotients/powers have laws.

Related Topics

Prerequisites

• Variables and Expressions
• Linear Equations

Related Topics

• Functions
• Quadratic Equations
• Algebra overview

Next Topics

• Limits
• Derivatives